Communications in Analysis and Geometry

Volume 31 (2023)

Number 1

Asymptotic convergence for modified scalar curvature flow

Pages: 69 – 96



Ling Xiao (Department of Mathematics, University of Connecticut, Storrs, Ct., U.S.A.)


In this paper, we study the flow of closed, star-shaped hypersurfaces in $\mathbb{R}^{n+1}$ with speed $r^\alpha \sigma^{1/2}_{2}$, where $\sigma^{1/2}_{2}$ is the normalized square root of the scalar curvature, $\alpha \geq 2$, and $r$ is the distance from points on the hypersurface to the origin. We prove that the flow exists for all time and the star-shapedness is preserved. Moreover, after normalization, we show that the flow converges exponentially fast to a sphere centered at origin. When $\alpha \lt 2$, a counter-example is given for the above convergence.

Received 19 May 2019

Accepted 12 August 2020

Published 21 September 2023